Winning Concurrent Reachability Games Requires Doubly-Exponential Patience

—We exhibit a deterministic concurrent reachability game PURGATORYn with n non-terminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given < 1/2 must use non-zero behavior probabilities that are less than ( 2 /(1 − ))2n−2 . Also, even to achieve the value within say 1 − 2−n/2 , doubly exponentially small behavior probabilities in the number of positions must be used. This behavior is close to worst case: We show that for any such game and 0 < < 1/2, there is an optimal strategy with all non-zero behavior probabilities being at least 2O(n) . As a corollary to our results, we conclude that any (deterministic or nondeterministic) algorithm that given a concurrent reachability game explicitly manipulates optimal strategies for Player 1 represented in several standard ways (e.g., with binary representation of probabilities or as the uniform distribution over a multiset)...

• Type: Conference
• Year: 2009
• Where: LICS
• Authors: Kristoffer Arnsfelt Hansen, Michal Koucký, Peter Bro Miltersen
• Keywords: Computer Science, Concurrent Reachability Game, Deterministic Concurrent Reachability, LICS 2009, Non-zero Behavior Probabilities